Nordström's second gravitational theory

[Altabeh]

A sort of simple question (equation of motion)

Homework Statement

Hello everybody
I'm new to these forums, so wish to be clear enough in my first post to not ask again.

In the Nordström's second theory of gravitation, the field equation is $$\varphi \,\square \left( \varphi \right) =4\,\pi { \it GT}_{{m}}$$ where $$\square$$ is the D'Alembertian operator defined in the Minkowskian spacetime with metric (+,-,-,-), T_m is the trace of the material contribution to the total stress-energy-momentum tensor $$T_{{\mu \nu }}$$ and finally we have $$\varphi$$ implying the potential.

This field is said to have the following Lagrangian proposed by Einstein: $$L={\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi }}-\rho \varphi$$ where $$\rho=\varphi \,T_{{m}}$$ is the density of matter.

Now my question is that how can one proceed to use the above Lagrangian to show that the equation of motion of a test particle moving in the field under discussion is $$\varphi \,d_{{\tau}} \left( u_{{\mu}} \right) =-\partial _{{\mu}} \left( \varphi \right) -d_{{\tau}} \left( \varphi \right) u_{{\mu}}$$ where $$\tau$$ is the proper time, $$u_{{\mu}}$$ is the 4-velocity of the moving particle and $$d_{{\tau}}(..)$$ refers to the derivative of (..) with respect to $$\tau$$??

Thanks in advance

AB

 

[NateD]

Homework Equations L={\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi }}-\rho \varphi \varphi \,d_{{\tau}} \left( u_{{\mu}} \right) =-\partial _{{\mu}} \left( \varphi \right) -d_{{\tau}} \left( \varphi \right) u_{{\mu}} The Attempt at a Solution To solve this problem, we will use the Euler-Lagrange equations. These equations are given by \frac{d}{{d\tau }} \left( \frac{{\partial L}}{{\partial \dot q}} \right)-\frac{{\partial L}}{{\partial q}} =0, where q is a generalized coordinate and \dot q is its derivative with respect to the proper time \tau . Now, in our case, the Lagrangian is given as L={\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi }}-\rho \varphi and the generalized coordinate is the 4-velocity of the particle, i.e. q=u_{{\mu}} . Thus, using the Euler-Lagrange equations, we obtain \frac{d}{{d\tau }} \left( \frac{{\partial L}}{{\partial \dot u_{{\mu}} }} \right)-\frac{{\partial L}}{{\partial u_{{\mu}} }} =0 or, \frac{d}{{d\tau }} \left( \frac{{\partial \left( {\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi